Abstract

We demonstrate the existence of the Ince–Gaussian beams that constitute the third complete family of exact and orthogonal solutions of the paraxial wave equation. Their transverse structure is described by the Ince polynomials and has an inherent elliptical symmetry. Ince–Gaussian beams constitute the exact and continuous transition modes between Laguerre and Hermite–Gaussian beams. The propagating characteristics are discussed as well.

© 2004 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 19.
  3. F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, 1964).
  4. F. M. Arscott, Proc. R. Soc. Edinburgh Sect. A 67, 265 (1967).
  5. I. Kimel and L. R. Elías, IEEE J. Quantum Electron. 29, 2562 (1993).
    [CrossRef]
  6. C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
    [CrossRef]
  7. S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
    [CrossRef]
  8. S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
    [CrossRef]
  9. J. Arlt and M. J. Padgett, Opt. Lett. 25, 191 (2000).
    [CrossRef]
  10. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  11. H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
    [CrossRef]

2003 (1)

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

2001 (1)

2000 (1)

1996 (1)

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

1993 (1)

I. Kimel and L. R. Elías, IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

1975 (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

1967 (1)

F. M. Arscott, Proc. R. Soc. Edinburgh Sect. A 67, 265 (1967).

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Arlt, J.

Arscott, F. M.

F. M. Arscott, Proc. R. Soc. Edinburgh Sect. A 67, 265 (1967).

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, 1964).

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

Chávez-Cerda, S.

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

Elías, L. R.

I. Kimel and L. R. Elías, IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

Gutiérrez-Vega, J. C.

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

Kimel, I.

I. Kimel and L. R. Elías, IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

McDonald, G. S.

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

New, G. H. C.

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

Okamoto, M.

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Padgett, M. J.

Sasada, H.

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

I. Kimel and L. R. Elías, IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

J. Math. Phys. (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

Opt. Commun. (1)

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Proc. R. Soc. Edinburgh Sect. A (1)

F. M. Arscott, Proc. R. Soc. Edinburgh Sect. A 67, 265 (1967).

Other (3)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 19.

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, 1964).

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Figures (3)

Fig. 1
Fig. 1

Transverse field distributions of some even IGBs with w0=3 and =2. Plots in the bottom row correspond to the phase structures of the modes displayed in the row immediately above them.

Fig. 2
Fig. 2

Same as Fig. 1 but with plots corresponding to odd IGBs.

Fig. 3
Fig. 3

Left, an IGB tends to a LGB or a HGB when 0 or , respectively. Right, reconstruction of the even IGB, IG5,3e=0.79LG0,5e0.35LG1,3e0.5LG2,1e.

Tables (1)

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Table 1 The Four Fundamental Modes

Equations (6)

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IGr=EξNηexpiZzΨGr,
d2Edξ2- sinh 2ξdEdξ-a-p cosh 2ξE=0,
d2Ndη2+ sin 2ηdNdη+a-p cos 2ηN=0,
-z2+zR2zRdZdz=p,
IGp,mer=Dw0wzCpmiξ,Cpmη,exp-r2w2z×expikz+ikr22Rz-ip+1arctanzzR,
LGn,lσIG¯p,mσdS=δσσδp,2n+l-1n+l+p+m/2×1+δ0,lΓn+l+1n!1/2×Al+δo,σ/2σapm,

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